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290 XIV INTEGRATION IN LOCALLY COMPACT GROUPS
which is (184.108.40.206). If both/and g are in Jfc(G)» then so is/*#; since
the formula (220.127.116.11) shows that the bilinear mapping (f,g)^f*g of
&l(G, j8) x &l(G, j?) into «C(G) is continuous (5.5.1), and since jfc(G) is
dense in &&G,P) and in ^c(G,j§) (13.11.6), the values of the mapping
(j9 #)h->/* # belong to the closure of Jf C(G) in #C(G) (3.11.4): that is, they
belong to <4(G) (13.20.5).
Proposition (14.10.7) implies the following corollary:
(14.10.8) Let A and B be two p-integrable sets in G. Then the function
XH-+j3(A n XB) w continuous on G tf/w/ tewcfe to 0 at infinity (13.20.6). IfE"1
is also p-integrable, the function jcn-»/?(A n xB) is fl-integrable, and we have
// moreover neither A nor B is ^negligible, the set AB * /z&s a nonempty
interior. Finally, for every subset A wWcA is ^-measurable and not ^negligible,
the set AA"1 is a neighborhood of the neutral element e ofG.
We have <pA e &2(G, ]8) and «pB e ^f2(G, j8), hence we may apply (14.10.7)
with /=<pA and # = <PB- Since ^B(5"Jx) = <pB(^"^) = ^B^). we have
(^A * $B)(X) = f ^A n xflW ^(^) = j5(A n ^B). This proves the first assertion.
If B"1 is also /Mntegrable, then both <pA and ^B-i belong to JSP^G, )8), and
we can apply (18.104.22.168), which gives us the formula j /?(A n xE) dft(x) =
]8(A)jS(B"1). If the right-hand side of this formula is nonzero, it follows
that /?(A n xB) is not identically zero, and hence there exists a nonempty open
set U in which the continuous function jS(A n *B) is >0. This implies that
U G AB""1. Finally, to prove the last assertion, we observe that there exists a
compact set K c A which is not ^-negligible, so that we may assume that A is
compact; but then j?(A) = (<pA * $A)(€) *s >^» an(^ ^ follows as above that
there exists a neighborhood of e contained in AA"1.
(14.10.9) Let /, g be two locally fi-integrable functions, (J. a measure on G.
The two following conditions are equivalent:
(a) fandg are convolvable and the function |/|* |^f| is p-integrable:
(b) ju and f are convolvable and the function g(\ji\ * |/|) is fi-mtegrable.
If these conditions are satisfied, we have
(22.214.171.124) <0, 0*(/'/9> = </*&/*>•
From (13.21.9) and (14.8.1), it follows that condition (a) means that thended.) Give an example in which the